An operational method for calculating the frequency fluctuations of a radio signal in a randomly inhomogeneous ionosphere

An operational method for calculating fluctuations of the Doppler frequency shift of a radio signal reflected from a randomly inhomogeneous ionosphere is proposed. The method is based on a numerical and analytical solutions of stochastic ray equations. The integral expressions are obtained for the average and root-mean-square deviations of the frequency of the radio signal along the oblique sounding path in the approximation of the perturbation method. The motion of chaotic ionospheric irregularities is taken into account in the framework of the hypothesis of the frozen turbulence transfer. The integral expressions for statistical moments are reduced to a system of ordinary differential equations of the first order and are solved numerically together with the ray equations in the regular ionosphere. This can significantly reduce the computer time spent calculating the Doppler frequency shift of the radio signal in a randomly inhomogeneous ionosphere. The results of mathematical modeling of frequency fluctuations of a decameter radio signal on a single-hop track in various geophysical conditions are presented.


Introduction
As is known [1], fluctuations of the Doppler frequency shift are observed. During the propagation of radio waves through an unsteady randomly inhomogeneous ionosphere with fixed coordinates of the source and receiver. Therefore, the operational assessment of the Doppler shift on operating ionospheric paths is very relevant for predicting the optimal conditions for the passage of a radio signal and improving its quality.
In general, the ionosphere route is a complex, multiply connected system [2,3]. The prediction of the characteristics of this system is a big problem due to the anisotropy of the ionosphere, the variety of types of ionospheric irregularities and the characteristics of radio wave propagation mechanisms. In particular, it is important to know the shape of the spectrum of ionospheric irregularities in estimating the frequency fluctuations of the radio signal. The phenomenon of birefringence occurs and the calculation of the characteristics of the radio signal for both magnetoion components is required due to the influence of the Earth's magnetic field on ionospheric radio paths. Despite these difficulties an IOP Publishing doi:10.1088/1742-6596/1847/1/012040 2 estimate of the frequency fluctuations of the radiosignal is possible on the basis of a model of the ionosphere with generalized (integral) properties. At present, the geophysical parameters of the thin structure of the ionosphere are known with a large degree of uncertainty [4], therefore, to predict the characteristics of the radio signal propagating in the channel, the radiophysical (effective) inhomogeneity parameters are also used, which are preliminarily obtained by approximate solution of the inverse problem from measurements of some characteristics of test signals on the reference tracks [5]. The radiophysical parameters of the irregularities obtained in this way are also of independent interest, since they contain integral information on the statistical variability of the ionosphere. Important results were obtained in this direction due to the introduction of ideas about a Gaussian correlation ellipsoid that effectively describes random ionospheric irregularities [6] and allows to significantly simplify the analytical calculations of the statistical moments of the radio signal. In the general case, the ionosphere is a multiscale medium and it is characterized by a powerlaw spectrum of irregularities. Nevertheless, the Gaussian spectrum of irregularities with effective parameters can be used when calculating the lowest moments of phase fluctuations of a radio signal. In particular, researches [7] showed that, when calculating the phase dispersion of a decameter radio signal in a multiscale randomly inhomogeneous ionosphere, a Gaussian model of the correlation ellipsoid can be used if the external scale of ionospheric turbulence specified by a powerlaw spectrum is taken as the spatial scale of the irregularities. It is connected to the fact that the highfrequency part of the spectrum of irregularities has a greater effect on the amplitude of the signal and to a lesser effect on its phase [8]. We consider the case of decameter radio signal propagation in the isotropic ionosphere in this paper. However, the proposed method for estimating the frequency fluctuations can also be used to calculate the Doppler frequency shift of individual magnetoionic components of the radio signal in the anisotropic ionosphere, if we take into account different refractive indices for the ordinary and extraordinary rays. The calculation method allows the introduction of an anisotropic correlation ellipsoid model of irregularities oriented relative to the radio path [9]. The parameters of this ellipsoid can be determined from the characteristics of the test signals on the reference paths, taking into account a priori information about the typical properties of irregularities (for example, their elongation along the lines of force of the geomagnetic field).
The aim of this work is to create an operational method for calculating the statistical characteristics of the Doppler frequency shift of a decameter radio signal with singlehop propagation in an unsteady randomly inhomogeneous ionosphere.

Derivation of analytic relations
By definition [10], Doppler frequency shift   of the received signal is the time  derivative of its phase  Using the approximation of geometric optics [11]: where  is frequency, c is speed of light in vacuum,  is random function of dielectric constant of plasma; integration is carried out along an arc S , connecting the receiving and emission points. Substituting (2) in (1) and following [1], we have We obtain the following expression, taking into account the curvature of the Earth's surface (figure 1) and equation (3): are beam path and angle of refraction, respectively; k  is angular coordinate of the radio reception point.

Figure 1. Coordinate system
We represent  in the form and performing asymptotic expansions: Integration is carried out along the average trajectory with characteristics in equations (6), (7). These characteristics can be determined by solving the system of ray equations in a plasma given by the average permittivity model We consider the motion velocities of chaotic inhomogeneities to be much greater than the velocity of the change in the average dielectric constant of the ionosphere: We have from (7) We use (11) to compile the statistical moment for the dispersion of the Doppler frequency shift of the radio signal: is spatiotemporal correlation function of the dielectric constant of the ionosphere irregularities, symbol means averaging over an ensemble of heterogeneities. Next, we consider a random field of irregularities that is quasihomogeneous in time and space. In this case, the function K has the form To describe the fluctuations of the dielectric constant of a randomly inhomogeneous ionosphere, correlation functions are usually used; the inverse Fourier transform of these functions gives a powerlaw spectrum of irregularitiess with a certain spectral index and parameters of the internal and external turbulence scales [9]. Meanwhile, the estimation of fluctuations of the Doppler frequency shift of the radio signal is also possible using the model of a Gaussian correlation ellipsoid of irregularities with generalized properties. Previously, such a model was effectively used in [6,13] to calculate the statistical moments of the angular characteristics of the radio signal reflected from the ionosphere. Studies [7,14] showed that when calculating the phase dispersion of a signal in a multiscale randomly inhomogeneous ionosphere, a Gaussian model of the correlation ellipsoid can be used, provided that the external scale of the plasma turbulence specified by the powerlaw spectrum is taken as the spatial scale of the irregularities. This possibility is explained by the fact that the highfrequency part of the irregularities spectrum has a greater effect on the amplitude of the radio signal and a lesser effect on its phase characteristics [8,9].
We define the homogeneous part of the correlation function 0 K in the form of a Gaussian function with the spatial correlation radius  , equal to the external turbulence scale of a random field of inhomogeneities. In this case, we obtain following expression by performing the sumdifference integration [8]  integration is carried out along the earth's surface. The following expression is obtained by assuming that the upper limit in the integral (15) is variable and differentiating the integral with respect to this limit We obtain a system of differential equations for simultaneously calculating the average trajectory, as well as the average and variance of the Doppler shift of the frequency of the radio signal, using equation (16) together with the system of deterministic ray equations (8), (9) and equation (6), differentiated by a variable upper limit . This system has the following form

Mathematical modelling and discussion of calculation results
Based on system (17), mathematical modeling of the influence of the random plasma irregularities drift on the width of the spectral line during oblique sounding of the ionosphere was carried out. The model of a single-layer deterministic ionosphere was considered: is critical frequency of layer;  The case of propagation of a decameter radio signal in a horizontally inhomogeneous ionosphere specified by a two-layer dielectric permittivity model was further considered. Layering violation is introduced using the model of horizontal large-scale heterogeneity. These changes were made to the dependence in the expression (18). As a result, a more general regular model of the ionosphere has the form: are respectively, the horizontal scale, the center coordinates and intensity of large-scale inhomogeneity. Figure 3 shows the of calculations results of the mean and standard deviation of the Doppler frequency shift of the radio signal in a horizontally inhomogeneous ionosphere (21). The function was given by the dependence: